A weaker weight condition for the John–Nirenberg implication

Let $v$ be a weight on the unit disk and let $p$ be the parameter appearing in Proposition 1. Assume that the function $x\mapsto v(1-x)x^{\frac{1}{p}-\epsilon}$ is almost increasing for some $\epsilon>0$. The weaker-weight conjecture. This assumption can be replaced by the milder condition

$$\inf_{x\in (0,1)}v(x)>0.$$

The proposed replacement would weaken the hypotheses of the proposition and potentially extend its John–Nirenberg-type conclusion to a broader class of weights.